RationalCubicSpline Class

public class RationalCubicSpline : CurveBase, 
	IInterpolationFunction, IInterpolationCurve

The basis functions for rational cubic splines are

  g1 = u
  g2 = t                     with   t = (x - x(i)) / (x(i+1) - x(i))
  g3 = u^3 / (p*t + 1)              u = 1 - t
  g4 = t^3 / (p*u + 1)

A rational spline with coefficients a(i),b(i),c(i),d(i) is determined by

         f(i)(x) = a(i)*g1 + b(i)*g2 + c(i)*g3 + d(i)*g4

Choosing the smoothing parameter p:
-----------------------------------

Use the method

     void MpRationalCubicSpline::SetSmoothing (double smoothing)

to set the value of the smoothing paramenter. A value of p = 0
for the smoothing parameter results in a standard cubic spline.
A value of p with -1 < p < 0 results in "unsmoothing" that means
overshooting oscillations. A value of p with p > 0 gives increasing
smoothness. p to infinity results in a linear interpolation. A value
smaller or equal to -1.0 leads to an error.


Choosing the boundary conditions:
---------------------------------

Use the method

     void MpRationalCubicSpline::SetBoundaryConditions (int boundary,
                      double b1, double b2)

to set the boundary conditions. The following values are possible:

     Natural
         natural boundaries, that means the 2nd derivatives are zero
         at both boundaries. This is the default value.

     FiniteDifferences
         use  finite difference approximation for 1st derivatives.

     Supply1stDerivative
         user supplied values for 1st derivatives are given in b1 and b2
         i.e. f'(x_lo) in b1
              f'(x_hi) in b2

     Supply2ndDerivative
         user supplied values for 2nd derivatives are given in b1 and b2
         i.e. f''(x_lo) in b1
              f''(x_hi) in b2

     Periodic
         periodic boundary conditions for periodic curves or functions.
         NOT YET IMPLEMENTED IN THIS VERSION.


If the parameters b1,b2 are omitted the default value is 0.0.


Input parameters:
-----------------

     Vector x(lo,hi)  The abscissa vector
     Vector y(lo,hi)  The ordinata vector
                      If the spline is not parametric then the
                      abscissa must be strictly monotone increasing
                      or decreasing!


References:
-----------
  Dr.rer.nat. Helmuth Spaeth,
  Spline-Algorithmen zur Konstruktion glatter Kurven und Flaechen,
  3. Auflage, R. Oldenburg Verlag, Muenchen, Wien, 1983.